A light aircraft pilot experienced a conflict with another aircraft as he crossed a runway. He thought he was cleared to cross; Local Control felt otherwise.

Narrative

I was returning to PDK (Atlanta/DeKalb Peachtree) on an instrument filed flight plan. I was cleared to land on Runway 20L and; upon landing; I believed my clearance from the Tower Controller to be exit right on Taxiway D; cross Runway 20R; hold on the other side and remain on the Tower Frequency; and this is the clearance I believe I read back. It is the clearance I wrote down on my kneeboard pad. Believing this to be my clearance; I exited Runway 20L to the right onto Taxiway D; and proceeded through the hold short line on Taxiway D between Runway 20L and Runway 20R toward the crossing of Taxiway D with Runway 20R. Upon taxiing through this hold short line; I did not receive any call or communication from the Tower directing me to stop; hold; or in any way indicating that I was violating an authorization or that I was not cleared to cross Runway 20R on Taxiway D; and this lack of any such com further reinforced my belief that I was cleared to cross Runway 20R. As I was crossing Runway 20R I saw another aircraft either landing or departing Runway 20R approaching my aircraft from the right. Again; I had not had any communication between the Tower Controller and the other aircraft indicating that a conflict existed between the aircraft or the clearance that had been provided. Inasmuch as I was already taxiing through Runway 20R when I first saw the approaching aircraft; upon seeing the approaching aircraft on Runway 20R I had no choice but to proceed on through Runway 20R on Taxiway D in order to avoid the prospect of conflict with the other aircraft. I did so and as I did; the other aircraft passed behind my aircraft on Runway 20R. At no time did I ever hear any communication between this aircraft and the Tower; not had the Tower communicated with me any further since my initial taxi clearance had been issued. I do not believe the other aircraft was at any time required to leave Runway 20R in order to avoid any conflict; and I was not required to leave Taxiway D in order to avoid any potential conflict. Upon reaching the hold short line on Taxiway D on the other side of Runway 20R I was; for the first time since the earlier clearance was issued; contacted by the Tower Controller and advised that an event that might require reporting had just occurred; given a telephone number and asked to call the Tower. I did so approximately 10 minutes later upon taxiing and exiting my aircraft.